There's surely no way they'd let a Klingon teenager on the bridge (if Klingons are adults by fourteen Earth-years old, you'd expect them to behave accordingly. Which isn't exactly "mature", but is at least not "teenager".)

Klingons have relatively long lifespans, though; if Kang and Koloth are any indication, 14 Klingon years would be about 30 Earth years. Reposition that into, say, a human being raised by someone like, say, the Salarians from Mass Effect who typically live about thirty to fourty years. A fourteen year old human might achieve a relatively high rank only because he has more life experience already than a middle-aged Salarian of superior rank. On the other hand, Klingon childhood and maturation is implied to be jaw-droppingly quick; Alexander gives a date for his birth as Stardate 43205, which corresponds to a couple of months AFTER Worf's holodeck bootycall in "The Emissary." It really sort of depends on which calendar we're using.

Either way, Worf's emotional immaturity is a matter of hormones and biochemical urges (he's horny all the time and he's ready for a fight) but he's dealing with those hormonal changes in the position of someone who has the life experience of a middle-aged human.

We don't really know how old Worf was when Khitomer was massacred, but he evidently remembers enough from his Klingon past -- and enough of Klingon culture -- that he didn't have to go off and re-connect with his heritage later in life. Seems the Rozhenkos raised him simply because too few of Mogh's living relatives survived the massacre to care for him or otherwise weren't contacted until Worf was already an adult... so I'm guessing he would have been about 5 or 10 Earth years at the time. In the Klingon lifecycle would have made him a grade schooler, but at that age he would have BEEN a grade schooler for almost the entirety of that decade before slowly growing towards puberty (which, curiously, he did not fully experience until Insurrection).

Memory Alpha has Worf being born in 2340; Khitomer was attacked in 2346, so he was five or six.

About what I figured. If Alexander is any indication of the Klingon lifecycle, Worf would have been the equivalent of a fourteen year old when he was adopted. That seems a bit awkward, except going by Worf's putative family history, his family always had a pretty close cultural/social relationship with humans anyway (much like Duras was close with the Romulans) and the Rozhenkos might have actually been family friends at the time.

Apparently he went back to Qo'nos at 15 do do his Klingon adulthood rites

Probably more like 8 or 10, about the same age Alexander was when he joined the Rotarran. Old enough to make a choice, but young enough that he still went back to Galt when it was over.